One can show that \(\mathcal {V}_\phi f\)  is uniformly continuous and moreover that \(\mathcal {V}_\phi f \in L^2(\mathbb {R}^2)\)  (see Lemma 3.1.1 and Theorem 3.2.1 in [1]}). Clearly, the Gabor transform — as defined in the introduction — corresponds to the short-time Fourier transform with window \(\phi = c \mathrm {e}^{-\pi (\cdot )^2}\) , where \(c = 2^{1/4}\) .
